Finite Group Databasebeta

← #6591325

Subgroups — #6591325

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36285C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic
4#33200C211yesmaximal, characteristic
5#33198C211yesmaximal, characteristic
6#28196C211yesmaximal, characteristic
7#28187C211yesmaximal, characteristic
8#27745C212yesmaximal
9#27742C212yesmaximal
10#26523C212yesmaximal
11#26521C212yesmaximal
12#24997C211yesmaximal, characteristic
13#24996C211yesmaximal, characteristic
14#24995C211yesmaximal, characteristic
15#24995C211yesmaximal, characteristic
16#24699C212yesmaximal
17#24691C212yesmaximal
18#23074C212yesmaximal
19#23073C212yesmaximal
20#22742C211yesmaximal, characteristic
21#22739C211yesmaximal, characteristic
22#21714C211yesmaximal, characteristic
23C4×#441C211yesmaximal, characteristic
24C2×#479C2212yes—
25C2×#477C2212yes—
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×#441C2211yescharacteristic
29C2×#441C2211yescharacteristic
30C2×C4×Q16C2211yescharacteristic
31#2868C2211yescharacteristic
32#2863—21no—
33#2862—21no—
34#2859C2211yescharacteristic
35#2606—22no—
36#2590—22no—
37#2582C2211yescharacteristic
38#2582C2211yescharacteristic
39#2582C2211yescharacteristic
40#2582C2211yescharacteristic
41#2571C2212yes—
42#2570—22no—
43#2570C2212yes—
44#2568C2212yes—
45#2565C2212yes—
46#2511—21no—
47#2511—21no—
48#2507—21no—
49#2446—22no—
50#2446—21no—
51#2370C2212yes—
52#2365C2212yes—
53#2356C2212yes—
54#2355C2212yes—
55#2355C2211yescharacteristic
56#2355C2211yescharacteristic
57#2355C2211yescharacteristic
58Q8×C4⋊C4C2211yescharacteristic
59C4×C4⋊Q8—21no—
60C4×C4⋊Q8C2211yescharacteristic
61#2295—21no—
62C42×Q8C2211yescharacteristic
63C42×Q8C2211yescharacteristic
64#2101C2212yes—
65#2101C2212yes—
66#2101C2212yes—
67#2100C2212yes—
68#2087C2212yes—
69#2086C2212yes—
70#2074C2212yes—
71#2074C2212yes—
72#2074C2212yes—
73#2072C2212yes—
74#2056C2212yes—
75#2056C2212yes—
76#1993C2212yes—
77#1993C2212yes—
78#1992C2212yes—
79#1992C2212yes—
80#1989C2212yes—
81#1987C2212yes—
82#1984C2211yescharacteristic
83#1984C2211yescharacteristic
84#1949C2212yes—
85#1949C2212yes—
86#1949C2212yes—
87#1949C2212yes—
88#1948C2212yes—
89#1948C2211yescharacteristic
90#1947C2212yes—
91#1947C2211yescharacteristic
92#1947C2211yescharacteristic
93#1947C2211yescharacteristic
94#1940C2212yes—
95#1940C2212yes—
96#1935C2212yes—
97#1935C2212yes—
98#1933C2212yes—
99#1933C2212yes—
100#1928C2212yes—
101#1928C2212yes—
102#1928C2212yes—
103#1927C2212yes—
104#1921C2212yes—
105#1919C2212yes—
106#1918C2212yes—
107#1918C2212yes—
108#1842C2211yescharacteristic
109#1836C2212yes—
110#1835C2212yes—
111#1834C2211yescharacteristic
112#1834C2211yescharacteristic
113#1834C2211yescharacteristic
114#1834C2211yescharacteristic
115#1805C2212yes—
116#1803C2211yescharacteristic
117#1803C2211yescharacteristic
118#1803C2211yescharacteristic
119#1803C2211yescharacteristic
120#1803C2212yes—
121#1776C2211yescharacteristic
122#1776C2211yescharacteristic
123#1776C2212yes—
124C4×#107C2212yes—
125C4×#106C2211yescharacteristic
126#1771C2212yes—
127#1771C2212yes—
128#1763C2211yescharacteristic
129#1763C2211yescharacteristic
130#1763C2211yescharacteristic
131#1763C2211yescharacteristic
132C4×#103C2211yescharacteristic
133C4×#103C2211yescharacteristic
134C4×#97C2211yescharacteristic
135C22×Q16C2311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8—21no—
138C2×C4⋊Q8D411yescharacteristic
139C2×C4⋊Q8C2312yes—
140C2×#125C2312yes—
141C2×C4×Q8—21no—
142C2×C4×Q8—22no—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148C2×C4×Q8C2311yescharacteristic
149C2×C4×Q8C2311yescharacteristic
150C2×C4×Q8C2311yescharacteristic
151#479—28no—
152#477—28no—
153#446—24no—
154C8×Q8—24no—
155#441—24no—
156#441—24no—
157C4×Q16—24no—
158C2×#107C2312yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#107C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2311yescharacteristic
164C2×#106C2311yescharacteristic
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2311yescharacteristic
176C2×#103C2311yescharacteristic
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#97C2312yes—
182C2×#97C2311yescharacteristic
183C2×#97C2311yescharacteristic
184C2×#97C2311yescharacteristic
185C2×#97C2311yescharacteristic
186C2×C4×C8C2311yescharacteristic
187C2×C4×C8C2312yes—
188C2×C4×C8C2311yescharacteristic
189#400—22no—
190#400—22no—
191#400—22no—
192#400—22no—
193#400—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#398—22no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—22no—
209#391—22no—
210#391—22no—
211#391D411yescharacteristic
212#391D411yescharacteristic
213#391C2312yes—
214#391C2312yes—
215#391C2312yes—
216#391C2312yes—
217#389—22no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389—21no—
222#389—22no—
223#389—21no—
224#389—22no—
225#389—21no—
226#389—21no—
227#389—21no—
228#389—21no—
229#389D412yes—
230#389D412yes—
231#389C2312yes—
232#389C2312yes—
233#389C2312yes—
234#389C2312yes—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#384—21no—
251#384—21no—
252#384D411yescharacteristic
253#384C2312yes—
254#384C2312yes—
255#384C2311yescharacteristic
256#384C2311yescharacteristic
257#384C2311yescharacteristic
258#384C2311yescharacteristic
259#383—22no—
260#382—22no—
261#382—22no—
262#382C2312yes—
263#382C2312yes—
264#382C2312yes—
265#382C2312yes—
266C4×C4⋊C4—22no—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—22no—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2312yes—
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2311yescharacteristic
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288C4×C4⋊C4C2312yes—
289#376—22no—
290C43—21no—
291C43—21no—
292C43C2311yescharacteristic
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2312yes—
302#340C2312yes—
303#340C2312yes—
304#340C2312yes—
305#340C2312yes—
306#340C2312yes—
307#340C2312yes—
308#340C2312yes—
309#340C2311yescharacteristic
310#340C2311yescharacteristic
311#340C2311yescharacteristic
312#340C2311yescharacteristic
313#340C2311yescharacteristic
314#340C2311yescharacteristic
315#336C2312yes—
316#336C2312yes—
317#336C2312yes—
318#336C2312yes—
319C22×Q8—22no—
320C22×Q8C2×D411yescharacteristic
321C22×Q8C2×D411yescharacteristic
322C22×Q8C2411yescharacteristic
323C22×Q8C2411yescharacteristic
324C2×Q16—22no—
325C2×Q16—22no—
326C2×Q16—22no—
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C22×C8C2412yes—
330C22×C8C2412yes—
331C22×C8C2411yescharacteristic
332C22×C8C2411yescharacteristic
333C4⋊Q8—24no—
334C4⋊Q8—24no—
335C4⋊Q8—42no—
336C4⋊Q8—44no—
337C4⋊Q8—82no—
338#125—24no—
339#125—24no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—22no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8—22no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—44no—
355C4×Q8—84no—
356C4×Q8#4114yes—
357C4×Q8#4114yes—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4C2×D411yescharacteristic
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D412yes—
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2×D412yes—
389C2×C4⋊C4C2×D412yes—
390C2×C4⋊C4C2×D412yes—
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2412yes—
398C2×C4⋊C4C2412yes—
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2411yescharacteristic
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—22no—
413C2×C42—22no—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—22no—
417C2×C42C2×D411yescharacteristic
418C2×C42C2×D411yescharacteristic
419C2×C42C2×D411yescharacteristic
420C2×C42C2×D411yescharacteristic
421C2×C42C2412yes—
422C2×C42C2412yes—
423C2×C42C2411yescharacteristic
424C2×C42C2411yescharacteristic
425C2×C42C2411yescharacteristic
426#107—22no—
427#107—24no—
428#107—24no—
429#107—24no—
430#106—22no—
431#106—22no—
432#106—24no—
433C4⋊C8—24no—
434C4⋊C8—24no—
435C4⋊C8—22no—
436C4⋊C8—22no—
437C4⋊C8—24no—
438C4⋊C8—24no—
439#103—22no—
440#103—22no—
441#103—24no—
442#103—24no—
443#103—24no—
444#103—24no—
445#103—22no—
446#103—22no—
447#103—22no—
448#103—22no—
449#97—24no—
450#97—22no—
451#97—22no—
452#97—22no—
453#97—22no—
454C4×C8—24no—
455C4×C8—22no—
456C4×C8#4114yes—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—22no—
470#95—22no—
471#95—22no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95—22no—
485#95—22no—
486#95—22no—
487#95C2×D412yes—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491C2×Q8—22no—
492C2×Q8—22no—
493C2×Q8—22no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—44no—
508C2×Q8—44no—
509C2×Q8—44no—
510C2×Q8C2×#4112yes—
511C2×Q8C2×#4112yes—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C2511yescharacteristic
535Q16—28no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—22no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8C2×#4112yes—
553C2×C8C2×#4112yes—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—24no—
557C4⋊C4—24no—
558C4⋊C4—24no—
559C4⋊C4—24no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—21no—
574C4⋊C4—22no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—22no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—42no—
582C4⋊C4—44no—
583C4⋊C4—44no—
584C4⋊C4—42no—
585C4⋊C4—42no—
586C4⋊C4—42no—
587C4⋊C4—42no—
588C4⋊C4—44no—
589C4⋊C4—44no—
590C4⋊C4—44no—
591C4⋊C4—42no—
592C4⋊C4—42no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—82no—
598C4⋊C4—82no—
599C4⋊C4—82no—
600C4⋊C4—82no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—81no—
604C4⋊C4—81no—
605C4⋊C4C2×#4112yes—
606C4⋊C4C2×#4112yes—
607C42—24no—
608C42—24no—
609C42—22no—
610C42—22no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—41no—
616C42—41no—
617C42—42no—
618C42—44no—
619C42—42no—
620C42—42no—
621C42—42no—
622C42—42no—
623C42—42no—
624C42—42no—
625C42—42no—
626C42—81no—
627C42—81no—
628C42—82no—
629C42—82no—
630C42—82no—
631C42C2×#4112yes—
632C42C2×#4112yes—
633C42C2×#4112yes—
634C422−1+412yes—
635C422−1+412yes—
636C422−1+412yes—
637C23C23×D411yescharacteristic
638Q8—22no—
639Q8—22no—
640Q8—22no—
641Q8—22no—
642Q8—44no—
643Q8—44no—
644Q8—41no—
645Q8—41no—
646Q8—41no—
647Q8—41no—
648Q8—88no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—22no—
674C2×C4—22no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—41no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—41no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—41no—
706C2×C4—42no—
707C2×C4#54712yes—
708C2×C4#54712yes—
709C2×C4#54812yes—
710C2×C4#54812yes—
711C2×C4#54812yes—
712C2×C4#54812yes—
713C2×C4#57512yes—
714C2×C4#57812yes—
715C2×C4C22×#4111yescharacteristic
716C2×C4C22×#4111yescharacteristic
717C2×C4C22×#4111yescharacteristic
718C2×C4C2×2−1+411yescharacteristic
719C2×C4C2×2−1+411yescharacteristic
720C2×C4C2×2−1+411yescharacteristic
721C8—22no—
722C8—22no—
723C8—42no—
724C8—42no—
725C8—41no—
726C8—41no—
727C22C2×#54711yescharacteristic
728C22C2×#54811yescharacteristic
729C22C2×#54811yescharacteristic
730C22C2×#57511yescharacteristic
731C22C2×#57811yescharacteristic
732C22#358511yescharacteristic
733C22#358611yescharacteristic
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—42no—
740C4—42no—
741C4—42no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—42no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—81no—
754C4—82no—
755C4#334914yes—
756C4#344411yescharacteristic
757C4#350911yescharacteristic
758C4#350911yescharacteristic
759C4#350911yescharacteristic
760C2#3615111yescharacteristic
761C2C2×#334911yescharacteristic
762C2#6129411yescharacteristic
763C2#6132011yescharacteristic
764C2#6132411yescharacteristic
765C2#6134211yescharacteristic
766C2#6134811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365982
#362852
#332972
#332002
#331982
#281962
#281872
#277452
#277422
#265232
#265212
#249972
#249962
#249952
#246992
#246912
#230742
#230732
#227422
#227392
#217142
C4×#4412